Exercise 10.1 (Disjunctive constraints)

Question

Suppose that we are given $m$ constraints $\mathbf{a}_i^{\prime} \mathbf{x} \geq b_i, i=1, \ldots, m$, but without the restriction $\mathbf{a}_i \geq 0$. Model the requirement that at least $k$ of them are satisfied. Assume that there exists a number $f$ such that $\mathbf{a}_i^{\prime} \mathbf{x} \geq f$ for $i=1, \ldots, m$, and for all feasible $\mathbf{x}$.

Ux Delta · Accepted Answer

Suppose $y_i\left\{\begin{array}{l}1, 	ext { if the ith constraint is satisfied } \0, 	ext { otherwise }\end{array}ight.$

The model can be formulated as follows:
$$\begin{align}
\mathbf{a}_i^{\prime} \mathbf{x} \geq b_iy_{i}+(1-y_{i})f\;\; &&i=1, \ldots, m \
\sum_{i=1}^{m}y_{i}\geq k  \
y_{i}\in \{0,1\}&&i=1,\dots,m
\end{align}$$
$f$ is a number that satisfies $\mathbf{a}_i^{\prime} \mathbf{x} \geq f$ for $i=1, \ldots, m$, and for all feasible $\mathbf{x}$.

Exercise 10.1 (Disjunctive constraints)

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Exercise 10.1 (Disjunctive constraints)

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